A binomial distribution applies when
1. Trials have exactly two possible outcomes
2. probabilities remain constant throughout trials.
3. there is a defined number of trials.
4. trials are independent of each other.
5. the random variable is the number of successes.
Since the problem appears to satisfy all the requirements, we will use a binomial distribution to model the probabilities.

p=P(A)=1/4
q=1-P(A)=3/4
From (p+q)^7, we look for the term
P(3)
=(7,3)p^3*q^4
=(7!/(3!4!))(1/4)^3(3/4)^4
=2835/16384